Implementing minimum cycle basis algorithms

Kurt Mehlhorn, Dimitrios Michail

Abstract: In this paper, we consider the problem of computing a minimum cycle basis of an undirected graph G = (V,E) with n vertices and m edges. We describe an efficient implementation of an O(m3 + mn2 log n) algorithm. For sparse graphs, this is the currently best-known algorithm. This algorithm's running time can be partitioned into two parts with time O(m3) and O(m2n + mn2 log n), respectively. Our experimental findings imply that for random graphs the true bottleneck of a sophisticated implementation is the O(m2 n + mn2 log n) part. A straightforward implementation would require Ω(nm) shortest-path computations. Thus, we develop several heuristics in order to get a practical algorithm. Our experiments show that in random graphs our techniques result in a significant speed-up. Based on our experimental observations, we combine the two fundamentally different approaches to compute a minimum cycle basis to obtain a new hybrid algorithm with running time O(m2n2). The hybrid algorithm is very efficient, in practice, for random dense unweighted graphs. Finally, we compare these two algorithms with a number of previous implementations for finding a minimum cycle basis of an undirected graph.